7 Answers
7

Lisp users refer to Lisp as the programmable programming language. It is used for symbolic computing - computing with symbols.

Macros are only one way to exploit the symbolic computing paradigm. The broader vision is that Lisp provides easy ways to describe symbolic expressions: mathematical terms, logic expressions, iteration statements, rules, constraint descriptions and more. Macros (transformations of Lisp source forms) are just one application of symbolic computing.

There are certain aspects to that: If you ask about 'redefining' the language, then redefine strictly would mean redefine some existing language mechanism (syntax, semantics, pragmatics). But there is also extension, embedding, removing of language features.

In the Lisp tradition there have been many attempts to provide these features. A Lisp dialect and a certain implementation may offer only a subset of them.

A few ways to redefine/change/extend functionality as provided by major Common Lisp implementations:

s-expression syntax. The syntax of s-expressions is not fixed. The reader (the function READ) uses so-called read tables to specify functions that will be executed when a character is read. One can modify and create read tables. This allows you for example to change the syntax of lists, symbols or other data objects. One can also introduce new syntax for new or existing data types (like hash-tables). It is also possible to replace the s-expression syntax completely and use a different parsing mechanism. If the new parser returns Lisp forms, there is no change needed for the Interpreter or Compiler. A typical example is a read macro that can read infix expressions. Within such a read macro, infix expressions and precedence rules for operators are being used. Read macros are different from ordinary macros: read macros work on the character level of the Lisp data syntax.

replacing functions. The top-level functions are bound to symbols. The user can change the this binding. Most implementations have a mechanism to allow this even for many built-in functions. If you want to provide an alternative to the built-in function ROOM, you could replace its definition. Some implementations will raise an error and then offer the option to continue with the change. Sometimes it is needed to unlock a package. This means that functions in general can be replaced with new definitions. There are limitations to that. One is that the compiler may inline functions in code. To see an effect then one needs to recompile the code that uses the changed code.

advising functions. Often one wants to add some behavior to functions. This is called 'advising' in the Lisp world. Many Common Lisp implementations will provide such a facility.

custom packages. Packages group the symbols in name spaces. The COMMON-LISP package is the home of all symbols that are part of the ANSI Common Lisp standard. The programmer can create new packages and import existing symbols. So you could use in your programs an EXTENDED-COMMON-LISP package that provides more or different facilities. Just by adding (IN-PACKAGE "EXTENDED-COMMON-LISP") you can start to develop using your own extended version of Common Lisp. Depending on the used namespace, the Lisp dialect you use may look slighty or even radically different. In Genera on the Lisp Machine there are several Lisp dialects side by side this way: ZetaLisp, CLtL1, ANSI Common Lisp and Symbolics Common Lisp.

CLOS and dynamic objects. The Common Lisp Object System comes with change built-in. The Meta-Object Protocol extends these capabilities. CLOS itself can be extended/redefined in CLOS. You want different inheritance. Write a method. You want different ways to store instances. Write a method. Slots should have more information. Provide a class for that. CLOS itself is designed such that it is able to implement a whole 'region' of different object-oriented programming languages. Typical examples are adding things like prototypes, integration with foreign object systems (like Objective C), adding persistance, ...

Lisp forms. The interpretation of Lisp forms can be redefined with macros. A macro can parse the source code it encloses and change it. There are various ways to control the transformation process. Complex macros use a code walker, which understands the syntax of Lisp forms and can apply transformations. Macros can be trivial, but can also get very complex like the LOOP or ITERATE macros. Other typical examples are macros for embedded SQL and embedded HTML generation. Macros can also used to move computation to compile time. Since the compiler is itself a Lisp program, arbitrary computation can be done during compilation. For example a Lisp macro could compute an optimized version of a formula if certain parameters are known during compilation.

Symbols. Common Lisp provides symbol macros. Symbol macros allow to change the meaning of symbols in source code. A typical example is this: (with-slots (foo) bar (+ foo 17)) Here the symbol FOO in the source enclosed with WITH-SLOTS will be replaced with a call (slot-value bar 'foo).

optimizations, with so-called compiler macros one can provide more efficient versions of some functionality. The compiler will use those compiler macros. This is an effective way for the user to program optimizations.

Condition Handling - handle conditions that result from using the programming language in a certain way. Common Lisp provides an advanced way to handle errors. The condition system can also be used to redefine language features. For example one could handle undefined function errors with a self-written autoload mechanism. Instead of landing in the debugger when an undefined function is seen by Lisp, the error handler could try to autoload the function and retry the operation after loading the necessary code.

Special variables - inject variable bindings into existing code. Many Lisp dialects, like Common Lisp, provide special/dynamic variables. Their value is looked up at runtime on the stack. This allows enclosing code to add variable bindings that influence existing code without changing it. A typical example is a variable like *standard-output*. One can rebind the variable and all output using this variable during the dynamic scope of the new binding will go to a new direction. Richard Stallman argued that this was very important for him that it was made default in Emacs Lisp (even though Stallman knew about lexical binding in Scheme and Common Lisp).

Lisp has these and more facilities, because it has been used to implement a lot of different languages and programming paradigms. A typical example is an embedded implementation of a logic language, say, Prolog. Lisp allows to describe Prolog terms with s-expressions and with a special compiler, the Prolog terms can be compiled to Lisp code. Sometimes the usual Prolog syntax is needed, then a parser will parse the typical Prolog terms into Lisp forms, which then will be compiled. Other examples for embedded languages are rule-based languages, mathematical expressions, SQL terms, inline Lisp assembler, HTML, XML and many more.

I'm going to pipe in that Scheme is different from Common Lisp when it comes to defining new syntax. It allows you to define templates using define-syntax which get applied to your source code wherever they are used. They look just like functions, only they run at compile time and transform the AST.

Here's an example of how let can be defined in terms of lambda. The line with let is the pattern to be matched, and the line with lambda is the resulting code template.

Note that this is NOTHING like textual substitution. You can actually redefine lambda and the above definition for let will still work, because it is using the definition of lambda in the environment where let was defined. Basically, it's powerful like macros but clean like functions.

Macros are the usual reason for saying this. The idea is that because code is just a data structure (a tree, more or less), you can write programs to generate this data structure. Everything you know about writing programs that generate and manipulate data structures, therefore, adds to your ability to code expressively.

Macros aren't quite a complete redefinition of the language, at least as far as I know (I'm actually a Schemer; I could be wrong), because there is a restriction. A macro can only take a single subtree of your code, and generate a single subtree to replace it. Therefore you can't write whole-program-transforming macros, as cool as that would be.

However, macros as they stand can still do a whole lot of stuff - definitely more than any other language will let you do. And if you're using static compilation, it wouldn't be hard at all to do a whole-program transformation, so the restriction is less of a big deal then.

This answer is specifically concerning Common Lisp (CL hereafter), although parts of the answer may be applicable to other languages in the lisp family.

Since CL uses S-expressions and (mostly) looks like a sequence of function applications, there's no obvious difference between built-ins and user code. The main difference is that "things the language provides" is available in a specific package within the coding environment.

With a bit of care, it is not hard to code replacements and use those instead.

Now, the "normal" reader (the part that reads source code and turns it into internal notation) expects the source code to be in a rather specific format (parenthesised S-expressions) but as the reader is driven by something called "read-tables" and these can be created and modified by the developer, it is also possible to change how the source code is supposed to look.

These two things should at least provide some rationale as to why Common Lisp can be considered a re-programmable programming language. I don't have a simple example at hand, but I do have a partial implementation of a translation of Common Lisp to Swedish (created for April 1st, a few years back).

I always thought it was because Lisp provided, at its core, such basic, atomic logical operators that any logical process can be built (and has been built and provided as toolsets and add-ins) from the basic components.

It is not so much that it can redefine itself as that its basic definition is so malleable that it can take any form and that no form is assumed/presumed into the structure.

As a metaphor, if you only have organic compounds you do organic chemistry, if you only have metal oxides you do metallurgy but if you have only elements you can do everything but you have extra initial steps to complete....most of which others have already done for you....

A reference to 'structure and interpretation of computer programs' chapter 4-5 is what I was missing from the answers (link).

These chapters guide you in building a Lisp evaluator in Lisp. I like the read because not only does it show how to redefine Lisp in a new evaluator, but also let you learn about the specifications of Lisp programming language.